Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
540 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#66330 | C2 | direct (N × H) |
| 6. C22×AΣL(1,𝔽27) | C4 | semidirect |
| 8. #66330 | C22 | direct (N × H) |
| 15. C2×AΣL(1,𝔽27) | C2×C4 | semidirect |
| 22. C22×C33 | Dic3 | semidirect |
| 25. C22×3−1+2 | Dic3 | semidirect |
| 27. C22×H(3,𝔽3) | Dic3 | semidirect |
| 31. AΣL(1,𝔽27) | C22×C4 | semidirect |
| 42. C2×C33 | C2×Dic3 | semidirect |
| 44. C2×3−1+2 | C2×Dic3 | semidirect |
| 46. C2×H(3,𝔽3) | C2×Dic3 | semidirect |
| 67. C33 | C22×Dic3 | semidirect |
| 68. 3−1+2 | C22×Dic3 | semidirect |
| 69. H(3,𝔽3) | C22×Dic3 | semidirect |
| 116. C22 | #66330 | direct (N × H) |
| 123. C2 | C2×#66330 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×AΣL(1,𝔽27) | 2 |
| C2×#66330 | 2 |
| C22×C3×C3⋊Dic3 | 3 |
| C22×3−1+2⋊C4 | 3 |
| C22×(C3⋊Dic3)⋊C3 | 3 |